export to obj - wrong material names and texture names
In Blender, a "Texture" is a slot that can contain a picture file (or something else, like a procedural, point cloud...). It is omitted when exporting, because in Obj a texture is a picture file. So your first material, mtl001, looks perfectly fine.

Are your sure your mtl002, mtl003 etc are actually assigned to some faces ? (In edit mode, select your material slot, select some faces, hit Assign under the slots list).

Try viewing your cube in Material shading instead of Texture shading, it will show you if the materials are correctly assigned.